Ricci flows with bursts of unbounded curvature

Abstract

Given a completely arbitrary surface, whether or not it has bounded curvature, or even whether or not it is complete, there exists an instantaneously complete Ricci flow evolution of that surface that exists for a specific amount of time [GT11]. In the case that the underlying Riemann surface supports a hyperbolic metric, this Ricci flow always exists for all time and converges (after scaling by a factor 1 2t ) to this hyperbolic metric [GT11], i.e. our Ricci flow geometrises the surface. In this paper we show that there exist complete, bounded curvature initial metrics, including those conformal to a hyperbolic metric, which have subsequent Ricci flows developing unbounded curvature at certain intermediate times. In particular, when coupled with the uniqueness from [Top13], we find that any complete Ricci flow starting with such initial metrics must develop unbounded curvature over some intermediate time interval, but that nevertheless, the curvature must later become bounded and the flow must achieve geometrisation as t → ∞, even though there are other conformal deformations to hyperbolic metrics that do not involve unbounded curvature.Another consequence of our constructions is that while our Ricci flow from [GT11] must agree initially with the classical flow of Hamilton and Shi in the special case that the initial surface is complete and of bounded curvature, by uniqueness, it is now clear that our flow lasts for a longer time interval in general, with Shi's flow stopping when the curvature blows up, but our flow continuing strictly beyond in these situations.All our constructions of unbounded curvature developing and then disappearing are in two dimensions. Generalisations to higher dimensions are then immediate. IntroductionHamilton [Ham82] and Shi [Shi89] proved that given a complete Riemannian manifold (M, g 0 ) with bounded curvature, there exists a complete Ricci flow g(t) on M for a short time, with g(0) = g 0 (see [Top06] for an introduction to this topic). The curvature of this Ricci flow is initially bounded, and the flow can be extended until such time that the curvature becomes unbounded.Ricci flows with possibly unbounded curvature in their initial condition and/or during the flow itself, were studied by the second author in [Top10] in the special case of surfaces, and in [GT11] we proved that one can always find an instantaneously complete Ricci flow starting at a completely arbitrary initial surface, whether of unbounded curvature or not, or indeed whether complete or not, which exists for a specific amount of time, and in [Top13] this solution was shown to be unique. More precisely, we proved: (1.1)Then there exists a smooth Ricci flow g(t) t∈[0,T ) such thatis complete for all t ∈ (0, T ));and this flow is unique in the sense that if g 2 (t) t∈[0,T2) is any other Ricci flow on M satisfying (i) and (ii), then T 2 ≤ T and g 2 (t) = g(t) for all t ∈ [0, T 2 ). If T < ∞, then we haveand in particular, T is the maximal existence time.It has been understood since the work of Hamilton and C...

Singular Ricci Solitons and Their Stability under the Ricci Flow

“…Finally, more results have been obtained in dimension 2 (see [16] for a survey): Giesen and Topping [14] (building on earlier work by Topping [20]) have given a construction of Ricci flows on surfaces starting from any (incomplete) initial metric whose curvature is unbounded; these solutions become instantaneously complete and are unique in the maximally stretched class that they introduce. More recently yet [15], they constructed examples of immortal solutions of the flow (on surfaces) which start out with a smooth initial metric, then the supremum of the Gauss curvature becomes infinite for some finite amount of time before becoming finite again. This paper considers a special class of singular initial metrics and produces examples of Ricci flow whose behavior is different from those listed above.…”

Applications of Hamilton’s compactness theorem for Ricci flow

“…It may be then hard to imagine how one could have T >T since this would imply that the curvature could blow up in our flow, but that we could then keep on flowing. However, this is exactly what can happen, for example: Theorem 5.3 (Proved with Giesen [15]). There exist a complete immortal Ricci flow g(t) t∈[0,∞) on C arising from Theorem 5.2, and a time t…”

“…For the two-dimensional theory of Ricci flow from the viewpoint of these lectures, one can see particularly [13,31,15,33], and we will also draw on elements of [12] and [14] in order to explain some subtleties of Hamilton's compactness theorem [32].…”

Short-time persistence of bounded curvature under the Ricci flow

“…Finally, we note that Cabezas-Rivas and Wilking [CW] and Giesen and Topping [GT2,GT3] have constructed examples which demonstrate that, when M is noncompact, the maximal time of existence of a smooth complete solution g(t) on M may strictly exceed the smallest T > 0 such that sup M×[0,T ) | Rm(g(t))| = ∞.…”